# Equivalence of two BV classes of functions in metric spaces, and existence of a Semmes family of curves under a $1$-Poincaré inequality

created by durandcar on 11 Sep 2018
modified by shanmugal on 22 Jan 2020

[BibTeX]

Accepted Paper

Inserted: 11 sep 2018
Last Updated: 22 jan 2020

Journal: Advances in Calculus of Variations
Year: 2019

Abstract:

We consider two notions of functions of bounded variation in complete metric measure spaces, one due to Martio and the other due to Miranda Jr. We show that these two notions coincide, if the measure is doubling and supports a $1$-Poincaré inequality. In doing so, we also prove that if the measure is doubling and supports a $1$-Poincaré inequality, then the metric space supports a Semmes family of curves structure.